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HOMOTOPY SIMPLEX PIVOT ALGORITHM
Xu Senlin, Liu Shulin
Acta mathematica scientia,Series B. 1991, 11 (1):
111-120.
In this paper, we study the properties of the zero set of a homotopy H:Im×[0, 1]→ Rm and its piecewise linear approximation φδi:Im×[0, 1]→Rm, These properties are very important for the homotopy simplex pivot algorithm. However, we prove that for almost every polynomial mapping the zero set of linear homotopy H(z, t)=tp(z)+(1-t)Q(z) consists of q=∏j=1nqj disjoint differential curves, and the zero set of its piecewise linear approximation φδi, consists of some broken lines. Where δi→0, these broken lines tend to differential curves in the zero set of H.
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